A Determinantal Formula for Catalan Tableaux and TASEP Probabilities

We present a determinantal formula for the steady state probability of each state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open boundaries, a 1D particle model that has been studied extensively and displays rich combinatorial structure. These steady state probabilities are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with $\alpha$'s and $\beta$'s that satisfy some conditions on the rows and columns. We construct a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram, and from this we enumerate the paths with a determinantal formula, building upon a formula of Narayana that counts unweighted lattice paths on a Young diagram. Finally, we provide a formula for the enumeration of Catalan tableaux that satisfy a given condition on the rows, which corresponds to the steady state probability that in the TASEP on a lattice with $n$ sites, precisely $k$ of the sites are occupied by particles. This formula is an $\alpha\ /\ \beta$ generalization of the Narayana numbers.Comment: 19 pages, 12 figure

Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex

The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of partially ordered sets on the entire collection of minors. We use the Lam and Postnikov circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset

Water hexamer: Self-consistent phonons versus reversible scaling versus replica exchange molecular dynamics

Classical free energies for the cage and prism isomers of water hexamer computed by the self- consistent phonons (SCP) method and reversible scaling (RS) method are presented for several flexible water potentials. Both methods have been augmented with a rotational correction for improved accuracy when working with clusters. Comparison of the SCP results with the RS results suggests a fairly broad temperature range over which the SCP approximation can be expected to give accurate results for systems of water clusters, and complements a previously reported assessment of SCP. Discrepancies between the SCP and RS results presented here, and recently published replica exchange molecular dynamics (REMD) results bring into question the convergence of the REMD and accompanying replica exchange path integral molecular dynamics results. In addition to the ever-present specter of unconverged results, several possible sources for the discrepancy are explored based on inherent characteristics of the methods used.Comment: Submitted to Journal Chemical Physic